Work Fluctuations in Slow Processes: Quantum Signatures and Optimal Control

An important result in classical stochastic thermodynamics is the work fluctuation-dissipation relation (FDR), which states that the dissipated work done along a slow process is proportional to the resulting work fluctuations. We show that slowly driven quantum systems violate this FDR whenever quantum coherence is generated along the protocol, and we derive a quantum generalization of the work FDR. The additional quantum terms in the FDR are found to lead to a non-Gaussian work distribution. Fundamentally, our result shows that quantum fluctuations prohibit finding slow protocols that minimize both dissipation and fluctuations simultaneously, in contrast to classical slow processes. Instead, we develop a quantum geometric framework to find processes with an optimal trade-off between the two quantities.

Figure 1

(a) Dissipated work $W_{\mathrm{diss}}$ and work fluctuations $\frac{1}{2}\beta {\sigma }_w^2$ as a function of initial state inverse temperature $\beta$ for the harmonic oscillator example. The plots are for a protocol in which the oscillator frequency ${\omega }_t$ is increased linearly in time from ${\omega }_0=0.1\tilde{\omega }$ to ${\omega }_1=10\tilde{\omega }$ for a fixed reference frequency $\tilde{\omega }$. (b) Plot of the metric tensors of fluctuations ($\mathrm{\Lambda }$) and dissipation ($\xi$), and of their difference ($\mathrm{\Lambda }-\xi$), for the harmonic oscillator example as a function of inverse temperature $\beta$ at a given energy gap $\hslash \omega$ (see Sec. F of the Supplemental Material [41]).

Figure 2

Pareto fronts limiting the accessible region of fluctuations $\frac{1}{2}\beta {\sigma }_w^2$ and dissipation $W_{\mathrm{diss}}$ for the harmonic oscillator example over all possible protocols $\{{\omega }_t\}$ between the end points ${\omega }_0=0.1\tilde{\omega }$ and ${\omega }_1=10\tilde{\omega }$ for a fixed reference frequency $\tilde{\omega }$. Curves are obtained by varying the weight $\alpha \in [0,1]$, and for each $\alpha$ by choosing the protocol to follow the geodesic that minimizes ${\mathcal{C}}_{\alpha }$. Each curve is for a specific inverse temperature $\beta =2\hslash \tilde{\omega }$ (blue), $\beta =1\hslash \tilde{\omega }$ (yellow), $\beta =0.7\hslash \tilde{\omega }$ (green), $\beta =0.6\hslash \tilde{\omega }$ (red), $\beta =0.5\hslash \tilde{\omega }$ (purple), $\beta =0.4\hslash \tilde{\omega }$ (brown), and $\beta =0.3\hslash \tilde{\omega }$ (light blue). The blue shaded region denotes the separation between the quantum optimal protocols (Pareto fronts) and the classical optimal protocols (diagonal) for varying $\beta$. (Inset) Magnified Pareto front for $\beta =\hslash \tilde{\omega }$ and including points for suboptimal protocols, illustrating the accessible part of the fluctuation-dissipation plane.